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This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman, and Tamminen [8], for target distributions with a non-compact support. The conditions ensuring a strong law of large numbers and a central limit theorem require that the tails of the target density decay super-exponentially, and have regular enough convex con...

The Metropolis-Hastings algorithm transforms a given stochastic matrix into a reversible stochastic matrix with a prescribed stationary distribution. We show that this transformation gives the minimum distance solution in an L1 metric.

Consider a d-ary rooted tree (d≥ 3) where each edge e is assigned an i.i.d. (bounded) random variable X(e) of negative mean. Assign to each vertex v the sum S(v) of X(e) over all edges connecting v to the root, and assume that the maximum S n of S(v) over all vertices v at distance n from the root tends to infinity (necessarily, linearly) as n tends to infinity. We analyze the Metropolis algori...

This paper considers high-dimensional Metropolis and Langevin algorithms in their initial transient phase. In stationarity, these algorithms are well-understood and it is now well-known how to scale their proposal distribution variances. For the random walk Metropolis algorithm, convergence during the transient phase is extremely regular to the extent that the algorithm’s sample path actually r...

Despite a few notable uses of simulation of random processes in the pre-computer era (Hammersley and Handscomb, 1964, Section 1.2; Stigler, 2002, Chapter 7), practical widespread use of simulation had to await the invention of computers. Almost as soon as computers were invented, they were used for simulation (Hammersley and Handscomb, 1964, Section 1.2). The name “Monte Carlo” started as cuten...

We investigate the hypothesis that the macroscopic properties of a porous material can be determined from limited morphological information. Specifically, we investigate this hypothesis for the Minkowski functionals of two-phase media in 2-D. We look at two methods for generating samples with desired Minkowski functionals: the Gibbs sampler and the Metropolis-Hastings algorithm. The Metropolis-...

This article contains an overview of the literature concerning the computational complexity of Metropolis-Hastings based MCMC methods for sampling probability measures on Rd , when the dimension d is large. The material is structured in three parts addressing, in turn, the following questions: (i) what are sensible assumptions to make on the family of probability measures indexed by d ?; (ii) w...

CPU Modeling and Refinement in Metropolis

Importance sampling and Metropolis-Hastings sampling (of which Gibbs sampling is a special case) are two methods commonly used to sample multi-variate probability distributions (that is, Bayesian networks). Heretofore, the sampling of Bayesian networks has been done on a conventional “classical computer”. In this paper, we propose methods for doing importance sampling and Metropolis-Hastings sa...